A shallow water model with eddy viscosity for basins with varying bottom topography
TL;DR: In this article, the authors introduce appropriate scalings into a three-dimensional anisotropic eddy viscosity model to derive a two-dimensional shallow water model and prove the global regularity of the resulting model.
Abstract: The motion of an incompressible fluid confined to a shallow basin with a varying bottom topography is considered. We introduce appropriate scalings into a three-dimensional anisotropic eddy viscosity model to derive a two-dimensional shallow water model. The global regularity of the resulting model is proved. The anisotropic form of the stress tensor in our three-dimensional eddy viscosity model plays a critical role in ensuring that the resulting shallow water model dissipates energy.
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TL;DR: In this article, the authors give some mathematical results for an isothermal model of capillary compressible fluids derived by Dunn and Serrin in 1985, which can be used as a phase transition model.
Abstract: In this article, we give some mathematical results for an isothermal model of capillary compressible fluids derived by Dunn and Serrin in [1]Dunn JE, Serrin J. On the thermodynamics of interstitial working. Arch Rational Mech Anal. 1985; 88(2):95–133), which can be used as a phase transition model. We consider a periodic domain Ω = T d (d = 2 ou 3) or a strip domain Ω = (0,1) × T d −1. We look at the dependence of the viscosity μ and the capillarity coefficient κwith respect to the density ρ. Depending on the cases we consider, different results are obtained. We prove for instance for a viscosity μ(ρ) = νρ and a surface tension the global existence of weak solutions of the Korteweg system without smallness assumption on the data. This model includes a shallow water model and a lubrication model. We discuss the validity of the result for the shallow water equations since the density is less regular than in the Korteweg case.
484 citations
Cites result from "A shallow water model with eddy vis..."
...We also refer to (2,14,17,22) for various results related to low Reynolds number regime, local in time results and semi-empirical derivation of the viscous stress....
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TL;DR: In this article, the authors consider a two dimensional viscous shallow water model with friction term and prove the existence of global weak solutions and convergence to the strong solution of the viscous quasi-geostrophic equation with free surface term.
Abstract: We consider a two dimensional viscous shallow water model with friction term. Existence of global weak solutions is obtained and convergence to the strong solution of the viscous quasi-geostrophic equation with free surface term is proven in the well prepared case. The ill prepared data case is also discussed.
444 citations
Cites background from "A shallow water model with eddy vis..."
...The reader is referred to [17] for an other viscous parametrization....
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TL;DR: In this article, the authors derived a two-dimensional viscous shallow water model in rotating framework, with irregular topography, linear and quadratic bottom friction terms and capillary effects.
Abstract: Resume Considering the three-dimensional Navier–Stokes equations with a free moving surface boundary condition and hydrostatic approximation, we study the derivation, with asymptotic analysis, of a new two-dimensional viscous shallow water model in rotating framework, with irregular topography, linear and quadratic bottom friction terms and capillary effects. A new formulation of the viscous effects, consistent with a previous one-dimensional analysis, is obtained. Finally, we propose some simple numerical experiments in order to validate the proposed model.
179 citations
TL;DR: In this article, the authors introduced appropriate scalings into a three-dimensional anisotropic eddy viscosity model; after averaging on the vertical direction and considering some asymptotic assumptions, they obtained a two-dimensional model, which approximates the threedimensional model at the second order with respect to the ratio between the vertical scale and the longitudinal scale.
Abstract: The motion of an incompressible fluid confined to a shallow basin with a slightly varying bottom topography is considered. Coriolis force, surface wind and pressure stresses, together with bottom and lateral friction stresses are taken into account. We introduce appropriate scalings into a three-dimensional anisotropic eddy viscosity model; after averaging on the vertical direction and considering some asymptotic assumptions, we obtain a two-dimensional model, which approximates the three-dimensional model at the second order with respect to the ratio between the vertical scale and the longitudinal scale. The derived model is shown to be symmetrizable through a suitable change of variables. Finally, we propose some numerical tests with the aim to validate the proposed model.
72 citations
Cites background or methods from "A shallow water model with eddy vis..."
...We also add a turbulence model for the stress tensor in NS3d, which we consider to present anisotropy with respect to the vertical and the longitudinal scale: therefore two different eddy viscosities (vertical and longitudinal) are introduced following [18]....
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...Following Levermore and Sammartino in [18], we observe that this relationship cannot be assumed to be isotropic because the horizontal and the vertical length scales will be of different orders in a shallow water approximation....
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...In [18] authors introduced a further eddy viscosity, say the bulk viscosity μe and they made the physical assumption that μe should be much smaller than μh; but they were dealing with the rigid lid approximation; in our case, without any rigid lid approximation, we take μh and μe of the same order of magnitude....
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01 Jan 2007
TL;DR: The purpose of this work is to present recent mathematical results about the shallow water model and to mention related open problems of high mathematical interest.
60 citations
References
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01 Jan 1967TL;DR: The dynamique des : fluides Reference Record created on 2005-11-18 is updated on 2016-08-08 and shows improvements in the quality of the data over the past decade.
Abstract: Preface Conventions and notation 1. The physical properties of fluids 2. Kinematics of the flow field 3. Equations governing the motion of a fluid 4. Flow of a uniform incompressible viscous fluid 5. Flow at large Reynolds number: effects of viscosity 6. Irrotational flow theory and its applications 7. Flow of effectively inviscid liquid with vorticity Appendices.
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01 Jan 1979TL;DR: This paper presents thediscretization of the Navier-Stokes Equations: General Stability and Convergence Theorems, and describes the development of the Curl Operator and its application to the Steady-State Naviers' Equations.
Abstract: I. The Steady-State Stokes Equations . 1. Some Function Spaces. 2. Existence and Uniqueness for the Stokes Equations. 3. Discretization of the Stokes Equations (I). 4. Discretization of the Stokes Equations (II). 5. Numerical Algorithms. 6. The Penalty Method. II. The Steady-State Navier-Stokes Equations . 1. Existence and Uniqueness Theorems. 2. Discrete Inequalities and Compactness Theorems. 3. Approximation of the Stationary Navier-Stokes Equations. 4. Bifurcation Theory and Non-Uniqueness Results. III. The Evolution Navier-Stokes Equations . 1. The Linear Case. 2. Compactness Theorems. 3. Existence and Uniqueness Theorems. (n < 4). 4. Alternate Proof of Existence by Semi-Discretization. 5. Discretization of the Navier-Stokes Equations: General Stability and Convergence Theorems. 6. Discretization of the Navier-Stokes Equations: Application of the General Results. 7. Approximation of the Navier-Stokes Equations by the Projection Method. 8. Approximation of the Navier-Stokes Equations by the Artificial Compressibility Method. Appendix I: Properties of the Curl Operator and Application to the Steady-State Navier-Stokes Equations. Appendix II. (by F. Thomasset): Implementation of Non-Conforming Linear Finite Elements. Comments.
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